It is easy to verify that a syntactic symmetry of a CNF formula is correct.
Is it also possible to check in polynomial time that a semantic symmetry which is not a syntactic symmetry of a formula is correct?
What is the complexity of the problem of checking the validity of a CNF formula's semantic symmetry?
Note that a syntactic (resp. semantic) symmetry $\sigma$ of a CNF formula $F$ is a permutation of its literals such that for all literal $x$, $\sigma(\neg x)=\neg \sigma(x)$ and $\sigma(F) = F$ (resp. $\sigma(F) \equiv F$).